Uniformly approachable maps∗

Throughout the paper we will use the standard definitions and notation ([ABR], [E]). Let X and Y be metric spaces. The goal is to study some intermediate classes of functions between the class Cuc(X,Y ) of all uniformly continuous mappings (briefly, UC) from X into Y and the class C(X,Y ) of all continuous functions f :X → Y . These classes, defined below, have been intensively studied in [BD] and [BDP] mainly in the case when Y = R. (Compare also [B, DP].) In this paper we will study them for general Y . In particular, we will consider the case when X = Y is the complex plane C. Let X and Y be metric spaces and let f :X → Y . 1. For K,M ⊆ X we say that g:X → Y is a (K,M)-approximation of f if g is a UC map such that g[M ] ⊆ f [M ] and g(x) = f(x) for each x ∈ K.